Download Calc 3.2 Guided Notes

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Calculus – 3.2 Guided Notes
Rolle’s Theorem and the Mean Value Theorem
The Extreme Value Theorem states that a continuous function on a closed interval [a, b] must have both
a minimum and a maximum on the interval.
Both of these values, however, can occur at the endpoints.
Rolle’s Theorem, named after the French mathematician Michel Rolle (1652–1719), gives conditions that
guarantee the existence of an extreme value in the interior of a closed interval.
EXAMPLE
Find the two x-intercepts of
f(x) = x2 – 3x + 2
and show that f’(x) = 0 at some point between the two x-intercepts.
Explain why Rolle’s Theorem does not apply for
f ( x) 
x
x2
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The Mean Value Theorem
Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) such that
Find a value c, such that f(c) would equal the average value on the interval.
a)
f ( x)  x3  2 x
[0, 2]
b)
f ( x)  2  x
[-7, 2]
Calculus – 3.2 Guided Notes